Pythagorean pants A comic name for the Pythagorean theorem, which arose due to the fact that those built on the sides of a rectangle and diverging in different sides the squares resemble the cut of pants. I loved geometry... and entrance exam to the university, I even received praise from Chumakov, a professor of mathematics, for the fact that without a board, drawing in the air with his hands, he explained the properties parallel lines and Pythagorean pants(N. Pirogov. Diary of an old doctor).

Phrasebook Russian literary language. - M.: Astrel, AST. A. I. Fedorov. 2008.

See what “Pythagorean pants” are in other dictionaries:

Pythagorean pants- ... Wikipedia

Pythagorean pants- Zharg. school Joking. Pythagorean theorem, establishing the relationship between the areas of squares built on the hypotenuse and legs right triangle. BTS, 835… Big dictionary Russian sayings

Pythagorean pants- A humorous name for the Pythagorean theorem, which establishes the relationship between the areas of squares built on the hypotenuse and legs of a right triangle, which looks like the cut of pants in the pictures... Dictionary of many expressions

Pythagorean pants (invent)- foreigner: about a gifted man Wed. This is undoubtedly a sage. In ancient times, he probably would have invented Pythagorean pants... Saltykov. Variegated letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse is equal to the squares of the legs (teaching ... ... Michelson's Large Explanatory and Phraseological Dictionary

Pythagorean pants are equal on all sides- The number of buttons is known. Why is the dick tight? (rudely) about pants and the male genital organ. Pythagorean pants are equal on all sides. To prove this, it is necessary to remove and show 1) about the Pythagorean theorem; 2) about wide pants... Live speech. Dictionary of colloquial expressions

Invent Pythagorean pants- Pythagorean pants (invent) monk. about a gifted person. Wed. This is undoubtedly a sage. In ancient times, he probably would have invented Pythagorean pants... Saltykov. Motley letters. Pythagorean trousers (geom.): in a rectangle there is a square of the hypotenuse... ... Michelson's Large Explanatory and Phraseological Dictionary (original spelling)

Pythagorean pants are equal in all directions- A humorous proof of the Pythagorean theorem; also as a joke about a friend's baggy trousers... Dictionary of folk phraseology

Adj., rude...

PYTHAGOREAN PANTS ARE EQUAL ON ALL SIDES (THE NUMBER OF BUTTONS IS KNOWN. WHY IS IT TIGHT? / TO PROVE THIS, YOU HAVE TO TAKE IT OFF AND SHOW)- adverb, rude... Dictionary modern colloquial phraseological units and proverbs

trousers- noun, plural, used compare often Morphology: pl. What? pants, (no) what? pants, what? pants, (see) what? pants, what? pants, what about? about pants 1. Pants are a piece of clothing that has two short or long legs and covers the lower part... ... Dmitriev's Explanatory Dictionary

Books

How the Earth was discovered, Sakharnov Svyatoslav Vladimirovich. How did the Phoenicians travel? What ships did the Vikings sail on? Who discovered America, and who made the first circumnavigation? Who compiled the world's first atlas of Antarctica, and who invented...

Famous Pythagorean theorem - "in a right triangle, the square of the hypotenuse equal to the sum squares of legs" - everyone knows it from school.

Well, do you remember "Pythagorean Pants", which "equal in all directions" - a schematic drawing explaining the theorem of the Greek scientist.

Here a And b - legs, and With - hypotenuse:

Now I will tell you about one original proof of this theorem, which you may not have known about...

But first let's look at one lemma - a proven statement that is useful not in itself, but for proving other statements (theorems).

Let's take a right triangle with vertices X, Y And Z, Where Z - a right angle and drop the perpendicular from right angle Z to the hypotenuse. Here W - the point at which the altitude intersects the hypotenuse.

This line (perpendicular) ZW splits the triangle into similar copies of itself.

Let me remind you that triangles are called similar, the angles of which are respectively equal, and the sides of one triangle are proportional to the similar sides of another triangle.

In our example, the resulting triangles XWZ And YWZ similar to each other and also similar to the original triangle XYZ.

This is not difficult to prove.

Let's start with triangle XWZ, note that ∠XWZ = 90, and therefore ∠XZW = 180–90-∠X. But 180–90-∠X - is exactly what ∠Y is, so triangle XWZ must be similar (all angles equal) to triangle XYZ. The same exercise can be done for the YWZ triangle.

The lemma is proven! In a right triangle, the altitude (perpendicular) dropped to the hypotenuse splits the triangle into two similar ones, which in turn are similar to the original triangle.

But, let’s return to our “Pythagorean pants”...

Drop the perpendicular to the hypotenuse c. As a result, we have two right triangles inside our right triangle. Let's label these triangles (in the picture above green) letters A And B, and the original triangle is a letter WITH.

Of course, the area of the triangle WITH equal to the sum of the areas of the triangles A And B.

Those. A+ B= WITH

Now let’s divide the figure at the top (“Pythagorean Pants”) into three house figures:

As we already know from the lemma, triangles A, B And C are similar to each other, therefore the resulting house figures are also similar and are scaled versions of each other.

This means that the area ratio A And a², - this is the same as the area ratio B And b², and also C And c².

Thus we have A/a² = B/b² = C/c² .

Let us denote this ratio of the areas of a triangle and a square in a house figure by the letter k.

Those. k - this is a certain coefficient that connects the area of the triangle (roof of the house) with the area of the square underneath it:
k = A / a² = B / b² = C / c²

It follows that the areas of the triangles can be expressed in terms of the areas of the squares under them in this way:
A = ka², B = kb², And C = kc²

But we remember that A+B = C, which means ka² + kb² = kc²

Or a² + b² = c²

And this is it proof of the Pythagorean theorem!

Some discussions amuse me immensely...

Hello, what are you doing?
-Yes, I’m solving problems from a magazine.
-Well, you give it to me! I didn't expect it from you.
-What didn’t you expect?
-That you will stoop to puzzles. You seem smart, but you believe in all sorts of nonsense.
-Sorry, I don't understand. What do you call nonsense?
-Yes, all this mathematics of yours. It’s obvious that it’s complete bullshit.
-How can you say that? Mathematics is the queen of sciences...
- Just let’s avoid this pathos, right? Mathematics is not a science at all, but one continuous pile of stupid laws and rules.
-What?!
-Oh, don’t make your eyes so big, you know yourself that I’m right. No, I don’t argue, the multiplication table is a great thing, it played a significant role in the formation of culture and human history. But now all this is no longer relevant! And then, why complicate everything? There are no integrals or logarithms in nature; these are all inventions of mathematicians.
-Wait. Mathematicians did not invent anything, they discovered new laws of interaction of numbers, using proven tools...
-Well, yes, of course! And do you believe this? Don't you see what nonsense they constantly talk about? Can you give me an example?
-Yes, please be kind.
-Yes please! Pythagorean theorem.
-Well, what’s wrong with it?
-It’s not like that! “Pythagorean pants are equal on all sides,” you understand. Did you know that the Greeks during the time of Pythagoras did not wear pants? How could Pythagoras even talk about something he had no idea about?
-Wait. What's this got to do with pants?
-Well, they seem to be Pythagoreans? Or not? Do you admit that Pythagoras didn't have pants?
- Well, actually, of course, it wasn’t...
-Aha, that means there is an obvious discrepancy in the very name of the theorem! How can you then take seriously what is said there?
- Just a minute. Pythagoras didn't say anything about pants...
-You admit it, right?
-Yes... So, can I continue? Pythagoras didn’t say anything about pants, and there’s no need to attribute other people’s stupidity to him...
-Yeah, you yourself agree that this is all nonsense!
-I didn’t say that!
-I just said that. You contradict yourself.
-So. Stop. What does the Pythagorean theorem say?
-That all pants are equal.
-Damn, did you even read this theorem?!
-I know.
-Where?
-I read.
-What did you read?!
-Lobachevsky.
*pause*
-Sorry, but what does Lobachevsky have to do with Pythagoras?
-Well, Lobachevsky is also a mathematician, and he seems to be an even greater authority than Pythagoras, wouldn’t you say?
*sigh*
-Well, what did Lobachevsky say about the Pythagorean theorem?
-That the pants are equal. But this is nonsense! How can you even wear such pants? And besides, Pythagoras didn’t wear pants at all!
-Lobachevsky said that?!
*second pause, with confidence*
-Yes!
-Show me where it is written.
-No, well, it’s not written so directly there...
-What is the name of the book?
- Yes, this is not a book, this is an article in a newspaper. About the fact that Lobachevsky was actually an agent German intelligence...well, that's beside the point. That's what he probably said anyway. He is also a mathematician, which means he and Pythagoras are at the same time.
-Pythagoras didn’t say anything about pants.
-Well, yes! That's what we're talking about. This is all bullshit.
-Let's go in order. How do you personally know what the Pythagorean theorem says?
-Oh, come on! Everyone knows this. Ask anyone, they will answer you right away.
-Pythagorean pants are not pants...
-Oh, of course! This is an allegory! Do you know how many times I've heard this before?
-The Pythagorean theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse. AND THAT'S ALL!
-Where are the pants?
-Yes, Pythagoras didn’t have any pants!!!
-Well, you see, that’s what I’m telling you. All your math is bullshit.
-But it’s not bullshit! See for yourself. Here's a triangle. Here's the hypotenuse. Here are the legs...
-Why suddenly these are the legs, and this is the hypotenuse? Maybe it's the other way around?
-No. Legs are two sides that form a right angle.
-Well, here's another right angle for you.
-He's not straight.
-What is he like, crooked?
-No, it's sharp.
-This one is also spicy.
-It's not sharp, it's straight.
-You know, don’t fool me! You just call things as it suits you, just to adjust the result to what you want.
-The two short sides of a right triangle are the legs. The long side is the hypotenuse.
-And who is shorter - that side? And the hypotenuse, therefore, no longer rolls? Listen to yourself from the outside, what kind of nonsense you are talking about. It’s the 21st century, the heyday of democracy, but you’re in some kind of Middle Ages. His sides, you see, are unequal...
-Rectangular triangle with equal sides doesn't exist...
-Are you sure? Let me draw it for you. Here, look. Rectangular? Rectangular. And all sides are equal!
-You drew a square.
-So what?
-A square is not a triangle.
-Oh, of course! As soon as it doesn’t suit us, it’s immediately “not a triangle”! Don't fool me. Count for yourself: one corner, two corners, three corners.
-Four.
-So what?
-It's a square.
-What is a square, not a triangle? He's worse, right? Just because I drew it? Are there three corners? There is, and there’s even one spare one. Well, there’s nothing wrong here, you know...
-Okay, let's leave this topic.
-Yeah, are you giving up already? Anything to object to? Do you admit that math is bullshit?
-No, I don’t admit it.
-Well, here we go again - great! I just proved everything to you in detail! If the basis of all your geometry is the teaching of Pythagoras, and, I apologize, it is complete nonsense... then what can we talk about further?
-The teachings of Pythagoras are not nonsense...
- Well, of course! I haven’t heard of the Pythagorean school! They, if you want to know, indulged in orgies!
-What does this have to do with...
-And Pythagoras was actually a fagot! He himself said that Plato was his friend.
-Pythagoras?!
-You didn’t know? Yes, they were all fagots. And kicked on the head. One slept in a barrel, the other ran around the city naked...
-Diogenes slept in a barrel, but he was a philosopher, not a mathematician...
-Oh, of course! If someone climbs into a barrel, then they are no longer a mathematician! Why do we need extra shame? We know, we know, we passed. But you explain to me why all sorts of fagots who lived three thousand years ago and ran around without pants should be an authority for me? Why on earth should I accept their point of view?
-Okay, leave it...
- No, listen! In the end, I listened to you too. These are your calculations, calculations... You all know how to count! And if I ask you something essentially, right there and then: “this is a quotient, this is a variable, and these are two unknowns.” And you tell me in general, without specifics! And without any unknown, unknown, existential... This makes me sick, you know?
-Understand.
-Well, explain to me why two and two are always four? Who came up with this? And why am I obliged to take it for granted and have no right to doubt?
- Yes, doubt it as much as you want...
-No, you explain to me! Only without these little things of yours, but normally, humanly, so that it’s clear.
-Twice two equals four, because two times two equals four.
-Oil oil. What new did you tell me?
-Twice two is two multiplied by two. Take two and two and put them together...
-So add or multiply?
-It's the same thing...
-Both-on! It turns out that if I add and multiply seven and eight, it also turns out the same thing?
-No.
-Why?
-Because seven plus eight does not equal...
-And if I multiply nine by two, do I get four?
-No.
-Why? I multiplied two and it worked, but suddenly it was a bummer with nine?
-Yes. Twice nine is eighteen.
-What about twice seven?
-Fourteen.
-And twice is five?
-Ten.
-That is, four turns out only in one particular case?
- That's right.
-Now think for yourself. You say that there are some strict laws and rules of multiplication. What kind of laws can we even talk about here if in each specific case a different result is obtained?!
-That's not entirely true. Sometimes the results may be the same. For example, twice six equals twelve. And four times three - too...
-Even worse! Two, six, three four - nothing in common at all! You can see for yourself that the result does not depend in any way on the initial data. The same decision is made in two radically different situations! And this despite the fact that the same two, which we take constantly and do not change for anything, always gives a different answer with all the numbers. Where, one wonders, is the logic?
-But this is just logical!
-For you - maybe. You mathematicians always believe in all kinds of crazy crap. But these calculations of yours do not convince me. And do you know why?
-Why?
-Because I I know, why your mathematics is actually needed. What does it all boil down to? “Katya has one apple in her pocket, and Misha has five. How many apples should Misha give to Katya so that they have the same number of apples?” And do you know what I'll tell you? Misha don't owe anyone anything give! Katya has one apple and that’s enough. Is she not enough? Let her work hard and honestly earn money for herself, even for apples, even for pears, even for pineapples in champagne. And if someone wants not to work, but only to solve problems, let him sit with his one apple and not show off!

PYTHAGOREAN PANTS ARE EQUAL ON ALL SIDES

This caustic remark (which in its entirety has a continuation: in order to prove it, you need to remove it and show it), invented by someone apparently shocked by the inner content of one important theorem of Euclidean geometry, reveals as accurately as possible the starting point from which the chain completely simple thought quickly leads to the proof of the theorem, as well as to even more significant results. This theorem, attributed to the ancient Greek mathematician Pythagoras of Samos (6th century BC), is known to almost every schoolchild and sounds like this: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. Perhaps many will agree that geometric figure, called the code “Pythagorean pants are equal on all sides,” is called a square. Well, with a smile on our face, let’s add a harmless joke for the sake of what was meant by the continuation of encrypted sarcasm. So, “to prove it, you need to film it and show it.” It is clear that “this” - the pronoun meant the theorem itself, “remove” - this means getting into your hands, taking the named figure, “show” - the word “touch” was meant, bringing some parts of the figure into contact. In general, “Pythagorean pants” was the name given to a graphic design resembling pants in appearance, which was obtained in Euclid’s drawing during his very complex proof of the Pythagorean theorem. When a simpler proof was found, perhaps some rhymer composed this tongue twister-hint so as not to forget the beginning of the approach to the proof, and popular rumor already spread it around the world as an empty saying. So, if you take a square, and place a smaller square inside it so that their centers coincide, and rotate the smaller square until its corners touch the sides of the larger square, then on the larger figure you will find 4 identical right triangles highlighted by the sides of the smaller square. From here there already lies a straight line the way to prove a famous theorem. Let the side of the smaller square be denoted by c. The side of the larger square is a+b, and then its area is (a+b) 2 =a 2 +2ab+b 2. The same area can be defined as the sum of the area of the smaller square and the areas of 4 identical right-angled triangles, that is, as 4 ab/2+c 2 =2ab+c 2. Let's put an equal sign between two calculations of the same area: a 2 +2ab+b 2 =2ab+c 2. After reducing the terms 2ab we get the conclusion: the square of the hypotenuse of a right triangle is equal to the sum squares of legs, that is, a 2 + b 2 =c 2. Not everyone will immediately understand the benefit of this theorem. From a practical point of view, its value lies in serving as a basis for many geometric calculations, such as determining the distance between points coordinate plane. Some valuable formulas are derived from the theorem; its generalizations lead to new theorems that bridge the gap from calculations in the plane to calculations in space. The consequences of the theorem penetrate into number theory, revealing individual details of the structure of a series of numbers. And much more, too many to list. A look from the point of view of idle curiosity demonstrates the presentation of entertaining problems by the theorem, which are formulated in an extremely clear way, but are sometimes tough nuts to crack. As an example, it is enough to cite the simplest of them, the so-called question of Pythagorean numbers, given in everyday terms as follows: is it possible to build a room, the length, width and diagonal on the floor of which would simultaneously be measured only in integer quantities, say in steps? Just the slightest change on this issue can make the task extremely difficult. And accordingly, there will be those who wish, purely out of scientific enthusiasm, to test themselves in splitting the next math puzzle. Another change to the question - and another puzzle. Often, in the search for answers to such problems, mathematics evolves, acquires fresh views on old concepts, and acquires new ones. systemic approaches and so on, which means the Pythagorean theorem, like any other worthwhile teaching, from this point of view is no less useful. Mathematics of the time of Pythagoras did not recognize numbers other than rational ones (natural numbers or fractions with a natural numerator and denominator). Everything was measured in whole quantities or parts of whole quantities. That’s why the desire to do geometric calculations and solve equations more and more is understandable. natural numbers. Addiction to them opens the way to incredible world the mysteries of numbers, a number of which are geometric interpretation initially appears as a straight line with infinite number marks Sometimes the dependence between some numbers in a series, the “linear distance” between them, the proportion immediately catches the eye, and sometimes the most complex mental constructions do not allow us to establish what patterns the distribution of certain numbers is subject to. It turns out that in the new world, in this “one-dimensional geometry”, the old problems remain valid, only their formulation changes. For example, a variant of the task about Pythagorean numbers: “From the house, the father takes x steps of x centimeters each, and then walks another steps of y centimeters. The son walks behind him z steps of z centimeters each. What should be the size of their steps so that at the z-th step did the child follow the father's trail?" To be fair, it should be noted that the Pythagorean method of developing thought is somewhat difficult for a novice mathematician. This is a special kind of style mathematical thinking, you need to get used to it. One interesting point. Mathematicians babylonian state(it arose long before the birth of Pythagoras, almost one and a half thousand years before him) also, apparently, knew some methods of searching for numbers, which later became known as Pythagorean. Cuneiform tablets were found where the Babylonian sages wrote down the triplets of such numbers that they identified. Some threesomes consisted of too many large numbers, in connection with which our contemporaries began to assume that the Babylonians had good, and probably even simple, methods for calculating them. Unfortunately, nothing is known about the methods themselves or their existence.

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Presentation on the topic: Pythagorean pants are equal in all directions

Slide no. 1

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Slide no. 3

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Perhaps many will agree that the geometric figure, called the code “Pythagorean pants are equal on all sides,” is called a square. Well, with a smile on our face, let’s add a harmless joke for the sake of what was meant by the continuation of encrypted sarcasm. So, “to prove it, you need to film it and show it.” It is clear that “this” - the pronoun meant the theorem itself, “remove” - this means getting into your hands, taking the named figure, “show” - the word “touch” was meant, bringing some parts of the figure into contact. In general, “Pythagorean pants” was the name given to a graphic design resembling pants in appearance, which was obtained in Euclid’s drawing during his very complex proof of the Pythagorean theorem. When a simpler proof was found, perhaps some rhymer composed this tongue twister-hint so as not to forget the beginning of the approach to the proof, and popular rumor already spread it around the world as an empty saying.

Slide no. 4

Slide description:

So, if you take a square, and place a smaller square inside it so that their centers coincide, and rotate the smaller square until its corners touch the sides of the larger square, then on the larger figure you will find 4 identical right triangles highlighted by the sides of the smaller square. From here there already lies a straight line the way to prove a famous theorem. Let the side of the smaller square be denoted by c. The side of the larger square is a+b, and then its area is (a+b) 2 =a 2 +2ab+b 2. The same area can be defined as the sum of the area of the smaller square and the areas of 4 identical right-angled triangles, that is, as 4 ab/2+c 2 =2ab+c 2. Let's put an equal sign between two calculations of the same area: a 2 +2ab+b 2 =2ab+c 2. After reducing the terms 2ab we get the conclusion: the square of the hypotenuse of a right triangle is equal to the sum squares of legs, that is, a 2 + b 2 =c 2.

Slide no. 5

Slide description:

Not everyone will immediately understand the benefit of this theorem. From a practical point of view, its value lies in serving as a basis for many geometric calculations, such as determining the distance between points on a coordinate plane. Some valuable formulas are derived from the theorem; its generalizations lead to new theorems that bridge the gap from calculations in the plane to calculations in space. The consequences of the theorem penetrate into number theory, revealing individual details of the structure of a series of numbers. And much more, too many to list.

Slide no. 6

Slide description:

A look from the point of view of idle curiosity demonstrates the presentation of entertaining problems by the theorem, which are formulated in an extremely clear way, but are sometimes tough nuts to crack. As an example, it is enough to cite the simplest of them, the so-called question about Pythagorean numbers, asked in everyday terms as follows: is it possible to build a room, the length, width and diagonal on the floor of which would simultaneously be measured only in integers, say in steps? Just the slightest change on this issue can make the task extremely difficult. And accordingly, there will be those who wish, purely out of scientific enthusiasm, to test themselves in cracking the next mathematical puzzle. Another change to the question - and another puzzle. Often, in the course of searching for answers to such problems, mathematics evolves, acquires fresh views on old concepts, acquires new systematic approaches, and so on, which means that the Pythagorean theorem, like any other worthwhile teaching, is no less useful from this point of view.

Slide no. 7

Slide description:

Mathematics of the time of Pythagoras did not recognize numbers other than rational ones (natural numbers or fractions with a natural numerator and denominator). Everything was measured in whole quantities or parts of whole quantities. That’s why the desire to do geometric calculations and solve equations more and more in natural numbers is so understandable. Addiction to them opens the way to the incredible world of the mystery of numbers, a number of which, in a geometric interpretation, initially appear as a straight line with an infinite number of marks. Sometimes the dependence between some numbers in a series, the “linear distance” between them, the proportion immediately catches the eye, and sometimes the most complex mental constructions do not allow us to establish what patterns the distribution of certain numbers is subject to. It turns out that in the new world, in this “one-dimensional geometry”, the old problems remain valid, only their formulation changes. For example, a variant of the task about Pythagorean numbers: “From the house, the father takes x steps of x centimeters each, and then walks another steps of y centimeters. The son walks behind him z steps of z centimeters each. What should be the size of their steps so that at the z-th step did the child follow the father's trail?"

Slide no. 8

Slide description:

To be fair, it should be noted that the Pythagorean method of developing thought is somewhat difficult for a novice mathematician. This is a special kind of style of mathematical thinking, you need to get used to it. One interesting point. The mathematicians of the Babylonian state (it arose long before the birth of Pythagoras, almost one and a half thousand years before him) also apparently knew some methods of searching for numbers, which later became known as Pythagorean numbers. Cuneiform tablets were found where the Babylonian sages wrote down the triplets of such numbers that they identified. Some triplets consisted of too large numbers, and therefore our contemporaries began to assume that the Babylonians had good, and probably even simple, methods for calculating them. Unfortunately, nothing is known about the methods themselves or their existence.

Pythagorean theorem pants. Interesting facts about the Pythagorean theorem: learn something new about the famous theorem

See what “Pythagorean pants” are in other dictionaries:

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Presentation on the topic: Pythagorean pants are equal in all directions